Method for decoding 2X2 spatiotemporal codes, in particular Golden type code

ABSTRACT

A method is provided for decoding spatiotemporal codes, in particular Golden type code. The received vector is subjected to a MMSE-GDFE filtering, a constellation re-centering to define a Z-matrix, a permutation to obtain X-shaped matrices, a trellis base reduction and a ZF-DFE algorithm by processing the elements of the Z-matrix layer by layer. Each layer includes both elements of the diagonal or anti-diagonal of the Z-matrix.

CROSS REFERENCE

The present application claims priority to PCT/FR2007/000798 filed May11, 2007.

FIELD OF THE INVENTION

The present invention relates to a method for decoding 2×2 space-timecodes of a vector Y, in particular of the Golden Code type. The presentinvention can also be applied to 2×2 space-time codes meeting thefollowing conditions:

-   -   the code has a structure in two layers,    -   each layer representing the canonical embedding of a principal        ideal of the ring of integers αZ[i,θ] of a number field Q[i,θ]        of degree 2 with respect to Q[i],    -   the canonical embedding matrix being proportional to a unitary        matrix.

BACKGROUND

In a general manner, in a wireless communication system, usingelectromagnetic waves as the physical transmission medium, multiplereflections (multiple paths) due to the different obstacles in thepropagation space (propagation in the urban environment or in the“indoor” environment) induce phenomena of fading of the signal, i.e.that over a complete frequency range, it is possible to observedestructive interference. The relative movement of the environment withrespect to the transmitter and to the receiver makes these fadingphenomena stochastic. These fading channels considerably penalise theperformance of conventional digital communication systems. To combatfading phenomena effectively, it is necessary that the receiver hasavailable several replicas containing the information transmitted,replicas which have suffered mutual independent fading. The concept ofdiversity refers to the creation of these replicas and the order ofdiversity is defined as the number of “independent” replicas which thereceiver has available. To mention some examples of diversity: timediversity (the information is repeated with a time difference greaterthan the coherence interval in order that there is decorrelation betweenthe replicas), frequency diversity, path diversity in CDMA systems,receive antenna diversity. Since 1996, a new coding method, namedSpace-time coding, using several transmit and receive antennas, hasallowed the use of a new form of diversity: transmit antenna diversity.Space-time coding is not the only technique resorting to severaltransmit and receive antennas: these techniques are combined under thename MIMO (Multiple Input Multiple Output). Recourse to Space-timecoding over several antennas considerably improves the performance(owing to diversity) and information rate (owing to multiplexing) oftelecommunications systems in an environment having fading (“fadingchannel”).

From all coherent Space-time codes (i.e. that the receiver has perfectknowledge or at least an estimate of the channel coefficients) using twotransmit antennas known at the present time, the one offering the bestperformance is the Golden Code as described in the document: J. C.Belfiore, G. Rekaya and E. Viterbo, “The Golden Code: A 2×2 Full-RateSpace-Time Code with Non-Vanishing Determinants,” IEEE Trans. Inform.Theory (April 2005). The Golden Code is actually a maximum diversity(the transmit diversity is 2) and maximum multiplexing (2 symbols perchannel use). Furthermore, it also has the highest coding gains(whatever the constellation used). In addition, it was demonstratedrecently that the Golden Code can be used on an “Amplify and ForwardRelay Channel” with a relay and an antenna and that it can benefit fromdiversity of collaboration.

The Golden Code only has excellent performance for Maximum Likelihooddecoding. As for a number of space-time codes (linear codes), theMaximum Likelihood decoding of the Golden Code is reduced to the searchwithin a region (the shaping region) of a lattice (of dimension 8 forthe Golden Code), a lattice which changes according to the channelcoefficients, from the point closest to the point received. In order todo this, algorithms to search for the closest point within a latticewere adapted to the decoding of Space-time codes: the Sphere Decoder andthe Schnorr-Euchner enumeration. These algorithms need a highly variablenumber of iterations and are therefore suited only with difficulty to ahardware implementation operating in real time. Accelerations of thesealgorithms using pre-processing, in particular basis reductions of theLLL type, have also been studied already.

Given the complexity of decoding at Maximum Likelihood decoding,sub-optimal algorithms for decoding Space-time codes have beendeveloped. Some of these algorithms stem from interference equalizationand cancellation: this is the case with the ZF (Zero Forcing), MMSE(Minimum Mean Square Error) and ZF-DFE (Decision Feedback Equalizer)algorithms described in: J. Foschini, G. Golden, R. Valenzuela and P.Wolniansky, “Simplified processing for high spectral efficiency wirelesscommunication employing multi-element arrays”, IEEE Journal on SelectedAreas Communications, vol. 17, p 1841-1852, November 1999, the V-BLASTdecoding algorithm. However, these decoders cannot benefit either fromtransmit diversity or receive diversity.

Recently, low complexity sub-optimal algorithms have been proposed. Theyuse MMSE or ZF algorithms combined with an LLL basis reduction. Thanksto LLL, these algorithms can use the diversity in its entirety, which isnot the case in the absence of this pre-processing. In the document: A.D. Murugan, H. El Gamal, M. O. Damen and G. Caire, “A Unified Frameworkfor Tree Search Decoding: Rediscovering the Sequential Decoder”, IEEETransactions on Information Theory, vol. 52, no. 5, May 2006, proved theinterest in using MMSE-GDFE filtering as pre-processing in order be ableto perform a decoding in the lattice without considering the “shaping”constraint and to make use reduction algorithms such as LLL. Moreover, ageneral formalising of searching within a tree was proposed, capableaccordingly of describing all the techniques resulting from the decodingwithin a lattice and introducing sequential decoding techniques.

However, even this latter technique cannot optimise the advantages ofthe Golden Code.

SUMMARY

A purpose of the present invention is to propose a new decoderbenefiting from the diversity of a Golden Code. Another purpose of theinvention is to design a real time space-time decoder.

At least one of the aforementioned objectives is achieved with a methodfor decoding 2×2 space-time codes of a vector Y, in particular of theGolden Code type, characterized in that it comprises the followingsteps:

-   -   a filtering of the MMSE-GDFE type is carried out in order to        obtain a vector Y^(F)=RX+W, where Y^(F) is the vector Y filtered        by means of a filter matrix F (“forward filter”), R a        conventional matrix named “backward filter”, X the vector        transmitted and W the complex noise matrix,    -   a constellation recentring is carried out so that the vector to        be determined becomes Z, such that        Z=Y^(F)−2K_(M)RΓS₀=2K_(M)RΓS+W, K_(M) being the normalization        constant for the constellation in question, Γ being the Golden        Code, and S the vector for the information symbols,    -   a step of permutation of the elements of the matrices Z, R and Γ        is carried out in order to obtain {tilde over (Z)}, {tilde over        (R)} and {tilde over (Γ)} so that {tilde over (R)}^(H)R is a        matrix in the form of an X and the complex number “i” is moved        from Γ to R, and    -   a step of lattice basis reduction is carried out so that {tilde        over (Z)}=2K_(M){tilde over (ψ)}{tilde over (Γ)}S′+W with {tilde        over (ψ)}={tilde over (R)}.Δ_(α)(x¹,x²,x³,x⁴) and S′=π_(α)        ⁻¹(x¹,x²,x³,x⁴).S, π_(α)(x¹,x²,x³,x⁴) being a basis change        matrix, π_(α)(x¹,x²,x³,x⁴) is the matrix associated with        Δ_(α)(x¹, x², x³, x⁴) which is a matrix in which the determinant        module is equal to 1 and defined such that Δ_(α) ^(H)Δ_(α) is a        matrix in the form of an X,    -   the ZF-DFE decoding algorithm is applied, in which among the        four elements of {tilde over (Z)} two layers are identified,        each layer comprising the two elements of the diagonal or        anti-diagonal of {tilde over (Z)}; the DFE algorithm is applied        layer by layer and the ZF algorithm within each of the layers.

DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

Further aspects of the present invention will become apparent from thefollowing description which is given by way of example only and withreference to the accompanying drawings in which:

FIG. 1 is a performance curve for a decoder according to the inventioncompared with a decoder of the prior art and a Maximum Likelihooddecoder; and

FIG. 2 is a flow chart of steps according to the invention.

DETAILED DESCRIPTION

The decoding algorithms of the prior art having performances comparableto Maximum Likelihood process the Golden Code decoding as that of anylattice of dimension 8. They do not consider two specific aspects of itsstructure: its construction in 2 layers and the algebraic structure ofeach of these two layers. The present invention proposes to carry out alayer by layer decoding (as in the method of a DFE), reducing theproblem to one of dimension 2. The decoding within each layer is carriedout by a simple ZF technique after an algebraic reduction. Such atechnique of decoding by layers has the advantage of handling at eachstep 2×2 diagonal, or known in advance, matrices. In order to obtainperformances close to Maximum Likelihood, a reduction capable ofcombining the two layers before decoding them one by one isadvantageously implemented. Conventional reduction techniques (LLL orKorkhine-Zolotareff, for example) destroy the layered structure. This iswhy a new form of reduction, christened algebraic reduction, can beused.

According to one advantageous characteristic of the invention, thepermutation of {tilde over (Z)} is as follows:

$\overset{\sim}{Z} = \begin{bmatrix}z_{1,1} \\z_{2,2} \\z_{1,2} \\z_{2,1}\end{bmatrix}$with the following convention

$Z = {\begin{bmatrix}z_{1,1} \\z_{2,1} \\z_{1,2} \\z_{2,2}\end{bmatrix}.}$

In this permutation, the first layer is constituted by the first andlast Z elements. The second layer is constituted by the second and thirdZ elements.

Advantageously, {tilde over (R)} is a matrix in X form such that:

${\overset{\sim}{R} = {{\begin{bmatrix}r_{1,1}^{1} & 0 & 0 & {i \cdot r_{1,2}^{1}} \\0 & r_{2,2}^{2} & r_{2,1}^{2} & 0 \\0 & r_{1,2}^{2} & r_{1,1}^{2} & 0 \\r_{2,1}^{1} & 0 & 0 & {i \cdot r_{2,2}^{1}}\end{bmatrix}\mspace{14mu}{and}\mspace{14mu}\overset{\sim}{\Gamma}} = {\frac{1}{\sqrt{5}}\begin{bmatrix}\alpha & {\alpha\;\theta} & 0 & 0 \\\overset{\_}{\alpha} & {\overset{\_}{\alpha}\;\overset{\_}{\theta}} & 0 & 0 \\0 & 0 & \alpha & {\alpha\theta} \\0 & 0 & \overset{\_}{\alpha} & {\overset{\_}{\alpha}\;\overset{\_}{\theta}}\end{bmatrix}}}},$α and θ being the Golden Code coefficients such that:

${\theta = \frac{1 + \sqrt{5}}{2}},{\overset{\_}{\theta} = \frac{1 - \sqrt{5}}{2}},$α=1+i−i.θ, α=1+i−i.{tilde over (θ)}

Preferably:

${\Delta_{\alpha}\left( {x^{1},x^{2},x^{3},x^{4}} \right)} = \begin{bmatrix}x^{1} & 0 & 0 & {\frac{\alpha}{\overset{\_}{\alpha}}x^{4}} \\0 & {\overset{\_}{x}}^{1} & {\frac{\overset{\_}{\alpha}}{\alpha}{\overset{\_}{x}}^{4}} & 0 \\0 & {\frac{\alpha}{\overset{\_}{\alpha}}{\overset{\_}{x}}^{2}} & {\overset{\_}{x}}^{3} & 0 \\{\frac{\overset{\_}{\alpha}}{\alpha}x^{2}} & 0 & 0 & x^{3}\end{bmatrix}$ and${\Pi_{\alpha}\left( {x^{1},x^{2},x^{3},x^{4}} \right)} = \begin{bmatrix}x_{1}^{1} & x_{2}^{1} & x_{1}^{4} & {x_{1}^{4} - x_{2}^{4}} \\x_{2}^{1} & {x_{1}^{1} + x_{2}^{1}} & x_{2}^{4} & {- x_{1}^{4}} \\{x_{1}^{2} + x_{2}^{2}} & {x_{1}^{2} + {2 \cdot x_{2}^{2}}} & {x_{1}^{3} + x_{2}^{3}} & {- x_{2}^{3}} \\{- x_{2}^{2}} & {{- x_{1}^{2}} - x_{2}^{2}} & {- x_{2}^{3}} & x_{1}^{3}\end{bmatrix}$

where ∀j 1≦j≦4 x^(j)=x₁ ^(j)+θ.x₂ ^(j) (x₁ ^(j) et x₂ ^(j)εZ[i]).

According to one advantageous embodiment of the invention, Δ_(α) isdetermined such that during a QR decomposition, both elements of asingle layer are substantially of the same value. This accordinglydefines a first equilibrium criterion within each layer.

According to yet another advantageous embodiment of the invention,{tilde over (Z)} being defined as two layers, a first layer composed ofelements on the diagonal and a second layer composed of elements on theanti-diagonal, Δ_(α) is determined such that, during a QR decomposition,the product of the two elements of the second layer is greater than theproduct of the two elements of the first layer. This accordingly definesa second criterion for maximizing the power of the second layer.

Moreover, during application of the ZF algorithm for the second layer,it is preferable to extract p possible candidates and then, duringapplication of the ZF algorithm for the first layer, p possiblecandidates are also extracted and then the best candidate vector isdetermined on completion of the overall ZF-DFE algorithm.Advantageously, p is equal to 2.

The present invention is applied advantageously if {tilde over (R)} andΔ_(α) are X or diamond matrices.

Other advantages and characteristics of the invention will becomeapparent on examination of the detailed description of an embodimentwhich is in no way limitative, and the attached figures. FIG. 1 is aperformance curve for a decoder according to the invention compared witha decoder of the prior art and a Maximum Likelihood decoder. FIG. 2 is aflow chart of steps according to the invention.

For the description which follows, we shall use the following notationsand conventions:

-   -   All assemblies in bold    -   Z the ring of relative integers    -   Q the field of rational numbers    -   R the field of real numbers        -   i is the number defined by i²=−1        -   C the field of complex numbers        -   Q[i] the field defined by Q[i]={x+i.y with (x,y)εQ²}        -   Z[i] the ring defined by Z[i]={x+i.y with (x, y)εZ²} (ring            of Gauss integers)        -   Q[θ] the algebraic extension field of Q generated by [θ]        -   Z[θ] the ring of algebraic integers from the number field            Q[θ]        -   θ is an algebraic integer of degree 2        -   Q[i,θ] the algebraic extension field of Q[i] generated by θ.            This is an algebraic extension of Q of degree 4.        -   Z[i,θ] the ring of algebraic integers of the body of numbers            Q[i,θ]        -   (x) the real part of a complex number x        -   (x) the imaginary part of a complex number x        -   x* is the complex conjugate of x        -   GL_(n)(E) all invertible square matrices of dimension n with            coefficients in E (where E is a ring) and in which the            inverse is also with coefficients in E        -   I_(n) the identity matrix of dimension n        -   X^(T) the transposed matrix of the X matrix        -   X^(H) the conjugated transposed matrix of the X matrix        -   └x┐ the largest integer less than or equal to x        -   In (x) the Napierian logarithm of x for x>0    -   E(.) the expected value.

The principle of the Golden Code will now be described.

For a constellation M² QAM, the Golden Code (annotated Γ_(M)) is thefinished set of the 2×2 matrices with complex coefficients defined asfollows:

$\Gamma_{M} = \left\{ {{{\frac{1}{\sqrt{5}}\begin{bmatrix}{\alpha \cdot \left( {a + {b \cdot \theta}} \right)} & {\alpha \cdot \left( {c + {d \cdot \theta}} \right)} \\{i \cdot \overset{\_}{\alpha} \cdot \left( {c + {d \cdot \overset{\_}{\theta}}} \right)} & {\overset{\_}{\alpha} \cdot \left( {a + {b \cdot \overset{\_}{\theta}}} \right)}\end{bmatrix}}{with}\mspace{14mu} a},b,c,{d \in {M^{2}{QAM}}}} \right\}$Where${\theta = \frac{1 + \sqrt{5}}{2}},{\overset{\_}{\theta} = \frac{1 - \sqrt{5}}{2}},{\alpha = {1 + i - {i \cdot \theta}}},{\overset{\_}{\alpha} = {1 + i - {i \cdot \overset{\_}{\theta}}}}$andM²QAM = {K_(M)((2 ⋅ k + 1 − M) + i ⋅ (2 ⋅ n + 1 − M))|(k, n) ∈ {0, …  , M − 1}²}where$K_{M} = {\frac{1}{\sqrt{\frac{2 \cdot \left( {M^{2} - 1} \right)}{3}}}\mspace{14mu}{is}\mspace{14mu}{the}\mspace{14mu}{normalization}\mspace{14mu}{constant}\mspace{14mu}{of}\mspace{14mu} M^{2}{{QAM}.}}$

In the MIMO context, the matrix notation is interpreted in the followingmanner: if a word X from the Golden Code (XεΓ_(M)) is transmitted, thismeans that:

-   -   x_(1,1) is sent by the antenna i=1 at time t=1.    -   x_(1,2) is sent by the antenna i=1 at time t=2.    -   x_(2,1) is sent by the antenna i=2 at time t=1.    -   x_(2,2) is sent by the antenna i=2 at time t=2.

By considering Γ_(M) in R⁸ (or C⁴), the Golden Code can be seen as aportion of a lattice of dimension 8 (or of complex dimension 4)decentred and delineated by a shaping region. To do this, it suffices torewrite Γ_(M) in the following manner:

$\Gamma_{M} = \begin{Bmatrix}{{\frac{2 \cdot K_{M}}{\sqrt{5}}\begin{bmatrix}{\alpha \cdot \left( {s_{1} + {s_{2} \cdot \theta}} \right)} & {\alpha \cdot \left( {s_{3} + {s_{4} \cdot \theta}} \right)} \\{i \cdot \overset{\_}{\alpha} \cdot \left( {s_{3} + {s_{4} \cdot \overset{\_}{\theta}}} \right)} & {\overset{\_}{\alpha} \cdot \left( {s_{1} + {s_{2} \cdot \overset{\_}{\theta}}} \right)}\end{bmatrix}} +} \\{\frac{K_{M} \cdot \left( {1 + i} \right) \cdot \left( {1 - M} \right)}{\sqrt{5}}\begin{bmatrix}{\alpha \cdot \left( {1 + \theta} \right)} & {\alpha \cdot \left( {1 + \theta} \right)} \\{i \cdot {\overset{\_}{\alpha}\left( {1 + \overset{\_}{\theta}} \right)}} & {\overset{\_}{\alpha} \cdot \left( {1 + \overset{\_}{\theta}} \right)}\end{bmatrix}} \\{{{avec}\left( {s_{1},s_{2},s_{3},s_{4}} \right)} \in {{Z\lbrack i\rbrack}^{4}\bigcap Q_{M}^{4}}}\end{Bmatrix}$

where Q_(M)=[0;M−1]+i.[0;M−1]

Subsequently, the information symbols vector is annotated S=(s₁, s₂, s₃,s₄)^(T). It is clear that the determination of the word of the GoldenCode is equivalent to the determination of the vector S of Z[1]⁴ in theshaping region Q_(M) ⁴.

Advantageously, a Golden Code word consists of two layers:

-   -   the pair (x_(1,1), x_(2,2)) forms the layer 1, which depends        only on the pair (s₁, s₂)    -   the pair (x_(1,2), x_(2,1)) forms the layer 2, which depends        only on the pair (s₃, s₄);

We repeat below the “idealistic” decoding criterion at MaximumLikelihood

Let us consider the case of a receiver having N receive antennas (N≧2).In the case of a MIMO Flat Fading channel (which is optionally differentbetween time 1 and time 2), if we suppose that Golden Code word X wastransmitted, the signals y_(j,t) received by the receive antenna j(1≦j≦N) at time t (1≦t≦2) can be written in the form:y _(j,t) =h _(j,1) ^(t) .x _(1,t) +h _(j,2) ^(t) x _(2,t) +b_(j,t)  (Equation 1)

where h_(j,i) ^(t) designates the channel coefficient between thetransmit antenna i and the receive antenna j at time t and b_(j,t) thecomplex noise on the receive antenna j at time t. The complex noise isassumed to be symmetric circularly centred Gaussian of variance N₀ (thereal and imaginary parts have for variance N₀/2) and independentaccording to space (according to j) and the time (according to t), i.e.:E(b _(j,t) b _(j′,t′)*)=N ₀δ_(t,t′)δ_(j,j′)

where δ_(t,t′) designates the Kronecker symbol between t and t′.

The channel coefficients and the noise variance are assumed to be knownby the receiver.

Equation (1) can be rewritten as a matrix:

$\underset{\underset{Y}{︸}}{\begin{bmatrix}y_{1,1} \\\vdots \\y_{N,1} \\y_{1,2} \\\vdots \\y_{N,2}\end{bmatrix}} = {{\underset{\underset{H}{︸}}{\begin{bmatrix}h_{1,1}^{1} & h_{1,2}^{1} & 0 & 0 \\\vdots & \vdots & \vdots & \vdots \\h_{N,1}^{1} & h_{N,2}^{1} & 0 & 0 \\0 & 0 & h_{1,1}^{2} & h_{1,2}^{2} \\\vdots & \vdots & \vdots & \vdots \\0 & 0 & h_{N,1}^{2} & h_{N,2}^{2}\end{bmatrix}} \cdot \begin{bmatrix}x_{1,1} \\x_{2,1} \\x_{1,2} \\x_{2,2}\end{bmatrix}} + \underset{\underset{B}{︸}}{\begin{bmatrix}b_{1,1} \\\vdots \\b_{N,1} \\b_{1,2} \\\vdots \\b_{N,2}\end{bmatrix}}}$

Now, given that XεΓ_(M), there is SεZ[i]⁴∩Q_(M) ⁴ such that:

$\begin{bmatrix}x_{1,1} \\x_{2,1} \\x_{1,2} \\x_{2,2}\end{bmatrix} = {{2 \cdot K_{M}}{\underset{\underset{\Gamma}{︸}}{\frac{1}{\sqrt{5}}\begin{bmatrix}\alpha & {\alpha\;\theta} & 0 & 0 \\0 & 0 & {i\;\overset{\_}{\alpha}} & {i\;\overset{\_}{\alpha}\;\overset{\_}{\theta}} \\0 & 0 & \alpha & {\alpha\;\theta} \\\overset{\_}{\alpha} & {{\overset{\_}{\alpha}\;\overset{\_}{\theta}}\;} & 0 & 0\end{bmatrix}} \cdot \left( {S + S_{0}} \right)}}$${{with}\mspace{14mu} S_{0}} = {\frac{\left( {1 + i} \right) \cdot \left( {1 - M} \right)}{2}\begin{bmatrix}1 \\1 \\1 \\1\end{bmatrix}}$

Therefore, we have:Y=2.K _(M) .H.Γ.(S+S ₀)+B

From which:Y′=Y−2.K _(M) .H.Γ.S ₀=2.K _(M) H.Γ.S+B

It is easy to demonstrate that the Maximum Likelihood decoding consistsof deciding, knowing Y′ and the channel coefficients, that the vectortransmitted is the vector Ŝ defined by:

$\hat{S} = {{Arg}\left( {\underset{S \in {{Z{\lbrack i\rbrack}}^{4}\bigcap Q_{M}^{4}}}{Min}{{{Y^{\prime\;} -} ⩓ {\cdot S}}}^{2}} \right)}$

with

=2.K_(M).H.Γ

Decoding at Maximum Likelihood consists of finding the point of thelattice of real dimension 8 (complex dimension 4), generated by

columns, closest to Y′, provided that the coordinates of this pointbelong to the shaping region (Q_(M) ⁴). Provided that the

matrix is of real rank 8 (or complex rank 4), a decoding at MaximumLikelihood is feasible by the Sphere Decoder algorithm or by theSchnorr-Euchner enumeration algorithm, both duly modified to integratethe “shaping” constraint.

According to the present invention, the first step is the application ofa MMSE-GDFE filter, 201 on FIG. 2.

The MMSE-GDFE filtering technique is a pre-processing technique usedcurrently in the Space-Time code decoding field. It has been shown thatMMSE-GDFE filtering can overcome the shaping constraint, withoutdegrading performance significantly. Moreover, this filtering canguarantee that the virtual channel matrix is of full rank. Bytransforming the decoding problem into that of a search within a latticefor the point closest to a point received, the MMSE-GDFE filtering canbenefit from lattice basis reduction techniques.

Given that the Golden Code matrix Γ is a unitary matrix, the MMSE-GDFEfiltering matrix can be determined exclusively from H and N₀ (bothassumed to be known by the receiver).

The MMSE-GDFE filter matrix, annotated F (“forward filter”), of thematrix H is the 4×2.N complex matrix defined by:

F=(R^(H))⁻¹H^(H), R is the backward filter,

where

$\left( R^{H} \right)^{- 1} = \begin{bmatrix}\left( R_{1}^{H} \right)^{- 1} & 0 \\0 & \left( R_{2}^{H} \right)^{- 1}\end{bmatrix}$with R₁ (or R₂) a triangular complex matrix greater than 2×2 (in whichthe diagonal terms are positive real numbers) obtained by the Choleskydecomposition of H^(1H)H¹+N₀I₂ (resp. H^(2H)H²+N₀I₂).

In general, decoding in a lattice after MMSE-GDFE filtering is definedin the following manner.

The received symbols vector Y is multiplied by the matrix F, in order toobtain a complex vector of dimension 4, annotated Y^(F):Y ^(F) =F.Y

It is easy to demonstrate that we have:Y ^(F) =R.X+W

where W is a complex noise vector of dimension 4, in which each of thecomponents is of variance N₀. The components of W are still mutuallydecorrelated, but are no longer Gaussian.

Therefore, we have, 202 on FIG. 2:Z=Y ^(F)−2.K _(M) .R.Γ.S ₀=2.K _(M) .R.Γ.S+W=

S+W  (Equation 2)

with

=2.K_(M).R.Γ

In this case, given Z and R, the decoding method consists of decidingthat the vector transmitted is the vector Ŝ defined by:

$\hat{S} = {{Arg}\left( {\underset{S \in {{Z{\lbrack i\rbrack}}^{4}\bigcap Q_{M}^{4}}}{Min}{{Z - {\Lambda^{F} \cdot S}}}^{2}} \right)}$

This decoding method is not equivalent to that of Maximum Likelihood,but damages performance only slightly while guaranteeing theinvertibility of the new generator matrix (

). Moreover, that it is not penalizing to disregard the shapingcondition. The decoding method then consists of deciding that the vectortransmitted is the vector Ŝ defined by:

$\hat{S} = {{Arg}\left( {\underset{S \in {Z{\lbrack i\rbrack}}^{4}}{Min}{{Z - {\Lambda^{F} \cdot S}}}^{2}} \right)}$

This therefore involves finding the point of the lattice generated bythe columns of

closest to Z.

If the coordinates of Ŝ do not confirm the “shaping” condition, it ispossible to decide to delete or to reduce Ŝ in the shaping region whilesaturating the coordinates of Ŝ which do not confirm the shapingcondition.

According to the invention, the decoder uses a lattice basis reduction.The general principle of lattice basis reduction is defined in thefollowing manner.

Let

be a square real matrix of dimension n and Z_(R) a point on R^(n).Suppose that we are searching for the point of the lattice generated by

closest to Z_(R), i.e. we are searching for Ŝ_(R) in Z^(n) such that:

${\hat{S}}_{R} = {{Arg}\left( {\underset{S_{R} \in Z^{n}}{Min}{{Z_{R} - {\Lambda_{R} \cdot S_{R}}}}^{2}} \right)}$

If π is a square matrix of dimension n with unimodular integercoefficients (πεGL_(n)(Z)), then its inverse π⁻¹ also has integercoefficients. The matrix π is called basis change matrix.

We therefore have:Z _(R) −

.S _(R) =Z _(R)−

.π.π⁻¹ .S _(R) =Z _(R) −

′.S _(R)′

with

′=

.π and S_(R)′=π⁻¹.S_(R)

Therefore the search for Ŝ_(R) is equivalent to the search for Ŝ′_(R)defined by:

${\hat{S}}_{R}^{\prime} = {{Arg}\left( {\underset{S_{R}^{\prime} \in Z^{n}}{Min}{{Z_{R} - {\bigwedge_{R}^{\prime}{\cdot S_{R}^{\prime}}}}}^{2}} \right)}$

Ŝ_(R) is obtained from the relationship: Ŝ_(R)=π.Ŝ_(R)′

The objective of the basis reduction is obtained from a

_(R) an equivalent generator matrix

_(R)′ (which generates the same lattice), which has improved vectororthogonality and length properties in order to facilitate the searchfor the closest point.

There are different reduction criteria, with which algorithms areassociated in order to carry out these reductions. The two principalreduction algorithms are: the LLL algorithm and the KZ algorithm. TheLLL algorithm is the more frequently used as it is less complex than theKZ algorithm.

The algorithms for classifying the columns of the matrix are alsoreduction algorithms.

After MMSE-GDFE filtering, the basis reduction can be used to acceleratethe optimal search algorithms such as the Sphere Decoder or theSchnorr-Euchner enumeration. Similarly, all the sub-optimal searchalgorithms in a tree can be improved using a basis reduction technique.

The general principle of a ZF decoding after MMSE-GDFE filtering andbasis reduction is as follows.

After MMSE-GDFE filtering, we have (cf. Equation 2):Z=

.S+W

In order to be able to use a reduction technique, it is necessary towork on real matrices and vectors. The conversion of a complex vector(resp. a complex matrix) into a real vector (resp. into a real matrix)is carried out in the following manner:

$Z_{R} = {{\begin{bmatrix} \\

\end{bmatrix}\mspace{14mu}{and}\mspace{14mu}\Lambda_{R}^{F}} = \begin{bmatrix} & - \\ & {\left( \Lambda^{F} \right)}\end{bmatrix}}$

In the case of the Golden Code, Z_(R) is a vector of dimension 8 and

an 8×8 matrix.

We then have:Z _(R) =

.S _(R) +W _(R)

After reduction of the generator matrix

, the reduced matrix is annotated

and the basis change matrix π.

We then have:Z _(R) =

S _(R) ′+W _(R)  (Equation 3)

The decoded word is defined by Ŝ_(R)=π.Ŝ_(R)′ where Ŝ_(R)′ is obtainedby rounding the coordinates of (

)⁻¹Z_(R). The term ZF (Zero Forcing) originates from the fact that wemultiply the received word Z_(R) by the inverse of the matrix

.

The general principle of a ZF-DFE decoding after MMSE-GDFE filtering andbasis reduction is as follows.

The ZF-DFE decoding after basis reduction proceeds like ZF decoding upto equation 3. A QR decomposition (or Gram-Shmit orthogonalisation) ofthe matrix

is carried out:

=U.T

with U a unitary real 8×8 matrix and T a real 8×8 upper triangularmatrix (in which the terms on the diagonal are positive). By multiplyingZ_(R) by U^(T), we obtain:Z _(R) ′=U ^(T) .Z _(R) =T.S _(R) ′+W _(R)′with

${E\left( {W_{R}^{\prime}W_{R}^{\prime\; T}} \right)} = {\frac{N_{0}}{2}I_{8}}$as U is unitary

The decoded word is defined Ŝ_(R)=π.Ŝ_(R)′ where Ŝ_(R)′ is obtainedcoordinate by coordinate in the following manner:

${\hat{s}}_{R,8}^{\prime} = \left\lfloor {\frac{z_{R,8}^{\prime}}{t_{8,8}} + 0.5} \right\rfloor$

For the coordinate 7, we refer the decision made on coordinate 8:

${\hat{s}}_{R,7}^{\prime} = \left\lfloor {\frac{z_{R,7}^{\prime} - {t_{7,8}{\hat{s}}_{R,8}^{\prime}}}{t_{7,7}} + 0.5} \right\rfloor$

and so on until:

${\hat{s}}_{R,1}^{\prime} = \left\lfloor {\frac{z_{R,1}^{\prime} - {\sum\limits_{j = 2}^{8}{t_{1,j}{\hat{s}}_{R,j}^{\prime}}}}{t_{1,1}} + 0.5} \right\rfloor$

We will now describe the decoding algorithm by layers according to thepresent invention and, in particular, its impact on the generalprinciple of the algorithms described above.

After MMSE-GDFE filtering, we carry out a permutation of Z, 203 on FIG.2. by re-ordering the lines of Z. We then define the vector {tilde over(Z)} in the following manner:

$\overset{\sim}{Z} = \begin{bmatrix}z_{1,1} \\z_{2,2} \\z_{1,2} \\z_{2,1}\end{bmatrix}$with the following convention

$Z = {\begin{bmatrix}z_{1,1} \\z_{2,1} \\z_{1,2} \\z_{2,2}\end{bmatrix}.}$

From equation (2), it is easy to show that:

$\overset{\sim}{Z} = {{2K_{M}\overset{\sim}{R}{\overset{\sim}{\Gamma} \cdot S}} + \overset{\sim}{W}}$with $\overset{\sim}{R} = \begin{bmatrix}r_{1,1}^{1} & 0 & 0 & {i \cdot r_{1,2}^{1}} \\0 & r_{2,2}^{2} & r_{2,1}^{2} & 0 \\0 & r_{1,2}^{2} & r_{1,1}^{2} & 0 \\r_{2,1}^{1} & 0 & 0 & {i \cdot r_{2,2}^{1}}\end{bmatrix}$ and$\overset{\sim}{\Gamma} = {{\frac{1}{\sqrt{5}}\begin{bmatrix}\alpha & {\alpha\;\theta} & 0 & 0 \\\overset{\_}{\alpha} & {\overset{\_}{\alpha}\overset{\_}{\theta}} & 0 & 0 \\0 & 0 & \alpha & {\alpha\;\theta} \\0 & 0 & \overset{\_}{\alpha} & {\overset{\_}{\alpha}\overset{\_}{\theta}}\end{bmatrix}} = {\frac{1}{\sqrt{5}}\begin{bmatrix}C_{\alpha,\theta} & 0_{2 \times 2} \\0_{2 \times 2} & C_{\alpha,\theta}\end{bmatrix}}}$

It will be noted that the matrix {tilde over (R)} is an X matrix. Wehave also moved “i” from {tilde over (Γ)} to {tilde over (R)}.

Some definitions and properties:

Definition 1: X matrix: a matrix will be said to be an X matrix if allthe coefficients outside the diagonal and the antidiagonal arenonexistent.

Proposition 1:

The product of two X matrices is an X matrix.

Definition 2:

Let (x^(j))_(1≦j≦4) be four elements of the ring of algebraic integersZ[i, θ] (where θ is an algebraic integer such that Q[i, θ] is analgebraic extension of degree 2 of Q[i]) and α an element of Z[i, θ], amatrix Δ_(α)(x¹, x², x³, x⁴) will be termed algebraic X if it iswritten:

${\Delta_{\alpha}\left( {x^{1},x^{2},x^{3},x^{4}} \right)} = \begin{bmatrix}x^{1} & 0 & 0 & {\frac{\alpha}{\overset{\_}{\alpha}}x^{4}} \\0 & {\overset{\_}{x}}^{1} & {\frac{\overset{\_}{\alpha}}{\alpha}{\overset{\_}{x}}^{4}} & 0 \\0 & {\frac{\alpha}{\overset{\_}{\alpha}}{\overset{\_}{x}}^{2}} & {\overset{\_}{x}}^{3} & 0 \\{\frac{\overset{\_}{\alpha}}{\alpha}x^{2}} & 0 & 0 & x^{3}\end{bmatrix}$where x is the algebraic conjugate of x.

Proposition 2:

Let Δ_(α)(x¹, x², x³, x⁴) be an algebraic X matrix, the determinant ofΔ_(α)(x¹, x², x³, x⁴) is given by:det(Δ_(α)(x ¹ ,x ² ,x ³ ,x ⁴))=N _(Q[i,θ]/Q[i])(x ¹ .x ³ −x ² .x ⁴)

where N_(Q[i,θ]/Q[i])(x)=x. x designates the algebraic norm of x (for xin Q[i,θ] relative to Q[i]. The determinant of Δ_(α)(x¹, x², x³, x⁴) isan element of Z[i] since it is the norm of the algebraic integerx¹x³−x²x⁴.

Proposition 3:

Let Δ_(α)(x¹, x² x³, x⁴) be an algebraic X matrix, then there exists aunique matrix π_(α)(x¹, x², x³, x⁴) with coefficients in Z[i] such that:

${{{\Delta_{\alpha}\left( {x^{1}, x^{2}, x^{3}, x^{4}} \right)} \cdot \left\lbrack \begin{matrix}C_{\alpha,\theta} & 0_{2 \times 2} \\0_{2 \times 2} & C_{\alpha,\theta}\end{matrix} \right\rbrack} = {\left\lbrack \begin{matrix}C_{\alpha,\theta} & 0_{2 \times 2} \\0_{2 \times 2} & C_{\alpha,\theta}\end{matrix} \right\rbrack \cdot {\Pi_{\alpha}\left( {x^{1}, x^{2}, x^{3}, x^{4}} \right)}}}$${{where}\mspace{14mu} C_{\alpha,\theta}} = \begin{bmatrix}\alpha & {\alpha\;\theta} \\\overset{\_}{\alpha} & {\overset{\_}{\alpha} \cdot \overset{\_}{\theta}}\end{bmatrix}$is the matrix of the canonical embedding of the ideal aZ[i,θ] in Z[i]

In the case of the Golden Code, the matrix π_(α)(x¹, x², x³, x⁴) isgiven by:

${\Pi_{\alpha}\left( {x^{1},x^{2},x^{3},x^{4}} \right)} = \begin{bmatrix}x_{1}^{1} & x_{2}^{1} & x_{1}^{4} & {x_{1}^{4} - x_{2}^{4}} \\x_{2}^{1} & {x_{1}^{1} + x_{2}^{1}} & x_{2}^{4} & {- x_{1}^{4}} \\{x_{1}^{2} + x_{2}^{2}} & {x_{1}^{2} + {2 \cdot x_{2}^{2}}} & {x_{1}^{3} + x_{2}^{3}} & {- x_{2}^{3}} \\{- x_{2}^{2}} & {{- x_{1}^{2}} - x_{2}^{2}} & {- x_{2}^{3}} & x_{1}^{3}\end{bmatrix}$where  ∀j  1 ≤ j ≤ 4  x^(j) = x₁^(j) + θ ⋅ x₂^(j)  (x₁^(j)  and  x₂^(j) ∈ Z[i])

Moreover, for the Golden Code, the matrix

$\frac{1}{\sqrt{5}}C_{\alpha,\theta}$is unitary.

Proposition 4:

Let Δ_(α)(x¹, x², x³, x⁴) be an algebraic X matrix and π_(α)(x¹,x², x³,x⁴) its associated matrix, then π_(α(x) ¹, x², x³, x⁴)εGL₄(Z[i]) if andonly if:|N _(Q[i,θ]/Q[i])((x ¹ .x ³ −x ² .x ⁴)|=1.

Similarly, we can define diamond matrices (nonexistant coefficients onthe diagonal and the antidiagonal) likely to obtain good results.

After permutation of Z, a basis reduction, 204 on FIG. 2, termedalgebraic reduction is carried out in the following manner.

Let Δ_(α)(x¹, x², x³, x⁴) be an algebraic X matrix with determinant ofmodule 1 and π_(α)(x¹,x²,x³,x⁴) its associated matrix. π_(α)(x¹,x², x³,x⁴) is an invertible matrix and its inverse has coefficients in Z[i].

We therefore have:{tilde over (Z)}=2.K _(M) {tilde over (R)}.{tilde over (Γ)}.S+{tildeover (W)}=2K _(M) {tilde over (R)}.Δ _(α() x ¹ ,x ² ,x ³ ,x ⁴).{tildeover (Γ)}.π_(α) ⁻¹ (x ¹ ,x ² ,x ³ ,x ⁴).S+{tilde over (W)}{tilde over (Z)}=2.K _(M) .{tilde over (ψ)}.{tilde over (Γ)}.S′+{tildeover (W)}

By having put: {tilde over (ψ)}={tilde over (R)}.Δ_(α)(x¹,x²,x³,x⁴) andS′=π_(α) ⁻¹(x¹,x²,x³,x⁴).S

The matrix {tilde over (ψ)} is an X matrix as well as a product of an Xmatrix. A reduction has therefore been carried out using the passagematrix π_(α)(x¹,x²,x³,x⁴) which has transformed the initial channelmatrix {tilde over (R)} into an equivalent channel matrix {tilde over(ψ)}.

The aim of the algebraic reduction is therefore to find a matrixΔ_(α)(x¹, x², x³, x⁴) so that the matrix {tilde over (ψ)} has improvedproperties to allow the use of a simple decoding technique while at thesame time obtaining good performance. To do this, two criteria aredefined: a balance within each layer and a maximization of the power ofthe second layer. These criteria will be described in more detail in thefollowing.

The sought properties of {tilde over (ψ)} therefore depend on thedecoding algorithm employed.

The decoding by layer will now be described for a number of candidates:using the ZF-DFE algorithms, 205 on FIG. 2.

The decoding principle for p candidates is as follows.

The decoding algorithm consists of determining p possible candidates forlayer 2. For each of the p layer 2 candidates, the closest value forlayer 1 is determined, while reporting the decision made on layer 2 (asfor the DFE method). Within each of the layers, the decoding techniqueis ZF made virtually optimal by algebraic reduction (step 2 of thereduction). The number of candidates, p, is a relatively low number: p=2is, for example, an excellent compromise between complexity andperformance. From the perspective of a real time system that is easy toimplement, it is preferable to fix the value of p. However, more complexstrategies for determining p, as a function of the matrix {tilde over(ψ)} could be envisaged.

-   -   The QR Decomposition

Let us suppose that the reduction has already been carried out; we thenhave:{tilde over (Z)}=2K _(M) .{tilde over (ψ)}.{tilde over (Γ)}.S′+{tildeover (W)}

A QR decomposition of the matrix {tilde over (ψ)} is carried out:

$\overset{\sim}{\Psi} = {\begin{bmatrix}\Psi_{1,1}^{1} & 0 & 0 & {i.\Psi_{1,2}^{1}} \\0 & \Psi_{2,2}^{2} & \Psi_{2,1}^{2} & 0 \\0 & \Psi_{1,2}^{2} & \Psi_{1,1}^{2} & 0 \\\Psi_{2,1}^{1} & 0 & 0 & {i.\Psi_{2,2}^{1}}\end{bmatrix} = {U \cdot T}}$

with U a unitary 4×4 complex matrix (U^(H)U=I₄) and T a upper triangular4×4 complex matrix in which the diagonal terms are real and positive. Inaddition, U and T have a particular structure:

$U = {{\begin{bmatrix}u_{1,1} & 0 & 0 & u_{1,4} \\0 & u_{2,2} & u_{2,3} & 0 \\0 & u_{3,2} & u_{3,3} & 0 \\u_{4,1} & 0 & 0 & u_{4,4}\end{bmatrix}\mspace{14mu}{and}\mspace{14mu} T} = \begin{bmatrix}t_{1,1} & 0 & 0 & t_{1,4} \\0 & t_{2,2} & t_{2,3} & 0 \\0 & 0 & t_{3,3} & 0 \\0 & 0 & 0 & t_{4,4}\end{bmatrix}}$

Given the form of {tilde over (ψ)}, its QR decomposition is equivalentto the QR decomposition of 2 2×2 matrices:

$\begin{bmatrix}\Psi_{1,1}^{1} & {i.\Psi_{1,2}^{1}} \\\Psi_{2,1}^{1} & {i.\Psi_{2,2}^{1}}\end{bmatrix} = {{\begin{bmatrix}u_{1,1} & u_{1,4} \\u_{4,1} & u_{4,4}\end{bmatrix} \cdot {\begin{bmatrix}t_{1,1} & t_{1,4} \\0 & t_{4,4}\end{bmatrix}\begin{bmatrix}\Psi_{2,2}^{2} & \Psi_{2,1}^{2} \\\Psi_{1,2}^{2} & \Psi_{1,1}^{2}\end{bmatrix}}} = {\begin{bmatrix}u_{2,2} & u_{2,3} \\u_{3,2} & u_{3,3}\end{bmatrix} \cdot \begin{bmatrix}t_{2,2} & t_{2,3} \\0 & t_{3,3}\end{bmatrix}}}$

We then have:{tilde over (Z)}=2K _(M) .U.T.{tilde over (Γ)}.S′+{tilde over (W)}

By multiplying by U^(H), we obtain:{tilde over (Z)}′=U ^(H) .{tilde over (Z)}=2K _(M) T.{tilde over(Γ)}.S′+{tilde over (W)}′

with E({tilde over (W)}′{tilde over (W)}′^(H))=N₀.I₄ since U is unitary.

-   -   The extraction of p candidates by ZF for the second layer.

The last two components of {tilde over (Z)}′ are:

$\begin{bmatrix}{\overset{\sim}{z}}_{3}^{\prime} \\{\overset{\sim}{z}}_{4}^{\prime}\end{bmatrix} = {{{2 \cdot K_{m} \cdot \begin{bmatrix}t_{3,3} & 0 \\0 & t_{4,4}\end{bmatrix} \cdot \frac{1}{\sqrt{5}}}{C_{\alpha,\theta} \cdot \begin{bmatrix}s_{3}^{\prime} \\s_{4}^{\prime}\end{bmatrix}}} + \begin{bmatrix}{\overset{\sim}{w}}_{3}^{\prime} \\{\overset{\sim}{w}}_{4}^{\prime}\end{bmatrix}}$

By applying a ZF (by multiplying by the inverse matrix), we obtain:

${\overset{\sim}{S}}_{2}^{\prime} = {\frac{1}{{2 \cdot K_{M}}\sqrt{5}} \cdot C_{\alpha,\theta}^{H} \cdot {\begin{bmatrix}{1/t_{3,3}} & 0 \\0 & {1/t_{4,4}}\end{bmatrix}\begin{bmatrix}{\overset{\sim}{z}}_{3}^{\prime} \\{\overset{\sim}{z}}_{4}^{\prime}\end{bmatrix}}}$

From which:

${\overset{\sim}{S}}_{2}^{\prime} = {\begin{bmatrix}s_{3}^{\prime} \\s_{4}^{\prime}\end{bmatrix} + {\frac{1}{{2 \cdot K_{M}}\sqrt{5}} \cdot C_{\alpha,\theta}^{H} \cdot \begin{bmatrix}{1/t_{3,3}} & 0 \\0 & {1/t_{4,4}}\end{bmatrix} \cdot \begin{bmatrix}{\overset{\sim}{w}}_{3}^{\prime} \\{\overset{\sim}{w}}_{4}^{\prime}\end{bmatrix}}}$

Starting from the vector {tilde over (S)}′₂, a list of distinct 2×2Z[i]² p vectors is established {Ŝ′₂ ¹, . . . , Ŝ′₂ ^(p)} closest to{tilde over (S)}′₂.

-   -   The ZF decoding of the first layer after referring p candidates

For each of p vectors Ŝ′₂ ^(j) (1≦j≦p), a vector Ŝ′₁ ^(j) is calculatedby rounding each of the coordinates of {tilde over (S)}′₁ ^(j) definedas follows:

${\overset{\sim}{S}}_{1}^{\prime\; j} = {\frac{1}{{2 \cdot K_{M}}\sqrt{5}} \cdot C_{\alpha,\theta}^{H} \cdot \begin{bmatrix}{1/t_{1,1}} & 0 \\0 & {1/t_{2,2}}\end{bmatrix} \cdot \begin{pmatrix}{\begin{bmatrix}{\overset{\sim}{z}}_{1}^{\prime} \\{\overset{\sim}{z}}_{2}^{\prime}\end{bmatrix} - {2 \cdot K_{M} \cdot}} \\\begin{matrix}{\begin{bmatrix}0 & t_{1,4} \\t_{2,3} & 0\end{bmatrix} \cdot} \\{\frac{1}{\sqrt{5}}{C_{\alpha,\theta} \cdot {\hat{S}}_{2}^{\prime\; j}}}\end{matrix}\end{pmatrix}}$

The vector Ŝ′^(j) is constructed:

${\hat{S}}^{\prime\; j} = \begin{bmatrix}{\hat{S}}_{1}^{\prime\; j} \\{\hat{S}}_{2}^{\prime\; j}\end{bmatrix}$

The vector Ŝ′^(j) is reduced in the initial base to obtain Ŝ^(j):Ŝ ^(j)=π_(α)(x ¹ ,x ² ,x ³ ,x ⁴).{circumflex over (S)}′^(j)

Thus a list of p distinct {Ŝ¹, . . . Ŝ^(p)} two by two Z[i]⁴ candidatevectors to be the decoded vector Ŝ is prepared.

-   -   determination of the best candidate.

Each of the vectors Ŝ^(j), is tested for membership of the shapingregion. In this way, we build a sub-list of the preceding list, whilekeeping only the vectors meeting the shaping condition. This list of kvectors (with 0≦k≦p) will be noted: {Ŝ^(σ(1)), . . . , Ŝ^(σ(k)}.) ⁾. Itis necessary to distinguish 2 cases:

-   -   k=0. Several strategies can be adopted without serious        consequences on performance. For example, we can choose to        saturate the coordinates of Ŝ′ which do not meet the shaping        condition, in order to obtain the decoded vectors Ŝ.    -   k≧1. In this case, the decoded vector Ŝ is defined as follows:

$\hat{S} = {{Arg}\left( {\underset{1 \leq j \leq k}{Min}{{\overset{\sim}{Z} - {{2 \cdot K_{M}}{\overset{\sim}{R} \cdot \overset{\sim}{\Gamma} \cdot {\hat{S}}^{\sigma{(j)}}}}}}^{2}} \right)}$

We will repeat below the criteria for {tilde over (ψ)} during thereduction algorithm.

-   -   equilibrium within each layer.

For each of the layers, a ZF decoding will be used. For the layer j(1≦j≦2), the 2×2 matrix A_(j) to be inverted is presented as follows:

$A_{j} = {{2 \cdot K_{M} \cdot \begin{bmatrix}t_{{{2 \cdot j} - 1},{{2 \cdot j} - 1}} & 0 \\0 & t_{{2 \cdot j},{2 \cdot j}}\end{bmatrix} \cdot \frac{1}{\sqrt{5}}}C_{\alpha,\theta}}$

Given that the matrix

$\frac{1}{\sqrt{5}}C_{\alpha,\theta}$is unitary, if t_(2.j-1, 2.j-1) and t_(2.j,2.j) are equal, then thematrix A_(j) is proportional to a unitary matrix, which would make theZF decoding equivalent to Maximum Likelihood within each of the layers.The aim of the reduction within layer j is therefore to equalize as faras possible the coefficients t_(2.j-1, 2.j-1) and t_(2.j,2.j).

-   -   to maximize the power of layer 2.

In order that the proposed decoder has the came behaviour as MaximumLikelihood, it is necessary that, from the p points chosen duringdecoding the second layer, one of them corresponds to that which theMaximum Likelihood decoder has decoded. To do this, it is necessary tomaximize the product t_(3,3).t_(4,4).

Given that the reduction does not modify the determinant module of T(i.e. the product t_(1,1).t_(2,2).t_(3,3).t_(4,4)), maximizing theproduct t_(3,3).t_(4,4) is equivalent to minimizing the productt_(1,1).t_(2,2).

It is important to note that:

$t_{1,1} = {{{\overset{\sim}{\Psi}}_{1}} = {{\sqrt{\sum\limits_{i = 1}^{4}{{\overset{\sim}{\Psi}}_{i,1}}^{2}}\mspace{14mu}{et}\mspace{14mu} t_{2,2}} = {{{\overset{\sim}{\Psi}}_{2}} = \sqrt{\sum\limits_{i = 1}^{4}{{\overset{\sim}{\Psi}}_{i,2}}^{2}}}}}$

An example of a reduction algorithm will now be described, taking intoaccount the two criteria of equilibrium and maximization.

Principle of the Reduction Algorithm:

The algebraic reduction algorithm is carried out in two steps:

-   -   The first step consists in determining a first quadruplet (β¹,        β², β³, β⁴) of algebraic integers which will allow minimization        of the product ∥{tilde over (φ)}₁∥.∥{tilde over (φ)}₂∥ where the        matrix {tilde over (φ)} is given by:        {tilde over (φ)}={tilde over (R)}.Δ _(α)(β¹,β²,β³,β⁴)    -   The second step consists of balancing each of the layers once        the first step is completed. This step does not modify the        balance between the layers, i.e. we will have        ∥{tilde over (φ)}₁∥.∥{tilde over (φ)}₂∥=∥{tilde over        (ψ)}₁∥.∥{tilde over (ψ)}₂∥

Preliminary Mathematics:

If we have: {tilde over (φ)}={tilde over (R)}.Δ_(α)(β¹,β²,β³,β⁴)

It will be noted that for a given matrix {tilde over (R)}:

-   -   ∥{tilde over (φ)}₁∥ depends only on (β¹,β²) which will be        denoted ∥{tilde over (φ)}₁∥(β¹,β²):

${{{\overset{\sim}{\Phi}}_{1}}\left( {\beta^{1},\beta^{2}} \right)} = \sqrt{{{{r_{1,1}^{1} \cdot \beta^{1}} + {{i.r_{{1,2}\;}^{1}}\frac{\overset{\_}{\alpha}}{\alpha}\beta^{2}}}}^{2} + {{{r_{2,1}^{1} \cdot \beta^{1}} + {{i.r_{2,2}^{1}}\frac{\overset{\_}{\alpha}}{\alpha}\beta^{2}}}}^{2}}$

-   -   ∥{tilde over (φ)}₂∥ depends only on ( β ¹, β ²):

${{{\overset{\sim}{\Phi}}_{2}}\left( {{\overset{\_}{\beta}}^{1},{\overset{\_}{\beta}}^{2}} \right)} = \sqrt{{{{r_{2,2}^{2} \cdot {\overset{\_}{\beta}}^{1}} + {r_{2,1}^{2}\frac{\alpha}{\overset{\_}{\alpha}}{\overset{\_}{\beta}}^{2}}}}^{2} + {{{r_{1,2}^{2} \cdot {\overset{\_}{\beta}}^{1}} + {r_{1,1}^{2}\frac{\alpha}{\overset{\_}{\alpha}}{\overset{\_}{\beta}}^{2}}}}^{2}}$

Definition:

A unit of a number field Q[i,θ] is an element of Z[i,θ] the inverse ofwhich is also in Z[i,θ]. Any unit of Q[i,θ] has an algebraic normrelative to a Q[i] which belongs to {1,−1, i, −i}.

Dirichlet's Theorem in a Real Quadratic Field:

Let Q[θ] be a totally real algebraic extension of Q of degree 2. Then aunit exists, called fundamental unit μ, such that any unit u of Q[θ] canbe written: u=(±1).μ^(k) with kεZ

Comment:

In the case of the Golden Code, we can take μ=θ

Proposition 5:

If the pair (β¹, β²) of algebraic integers from Z[i,θ] minimises theproduct ∥{tilde over (φ)}₁∥.∥{tilde over (φ)}₂∥, then β¹ and β² have nocommon multiples other than units of Z[i,θ].

Proof:

Let us suppose that (β¹, β²) minimises ∥{tilde over (φ)}₁∥.∥{tilde over(φ)}₂∥ and that xεZ[i,θ] exist such that β¹=x.γ¹ et β²=x.γ². Then∥{tilde over (φ)}₁∥(β¹,β²).∥{tilde over (φ)}₂∥( β ¹, β ²)=|N_(Q[i,θ]/Q[i])(x)∥{tilde over (φ)}₁∥(γ¹,γ²).∥{tilde over (φ)}₂∥( γ ¹, γ²)

from which |N_(Q[i,θ]/Q[i])(x)=1.

Therefore x is a unit of Z[i,θ].

Proposition 6:

If the ring of algebraic integers Z[i,θ] is a PID (Principal IntegralDomain), which is the case for the Golden Code, and if in addition β¹and β² are two algebraic integers of Z[i,θ], which have no commonmultiples other than units of Z[i,θ], then two algebraic integers β³ andβ⁴ exist such that:β¹.β³−β².β⁴=1

Proof:

From proposal 5, we use a generalisation of Bézout's theorem in a PID,or even in particular the document: H. Cohen, “A Course in ComputationalAlgebraic Number Theory”, Springer, 1996.

Proposition 7:

If the pair (β¹, β²) of algebraic integers of Z[i,θ] minimises theproduct ∥{tilde over (φ)}₁∥.∥{tilde over (φ)}₂∥ and if, in additionZ[i,θ] is a PID, then there are two algebraic integers β³ and β⁴ ofZ[i,θ] (which are determined by resolving β¹.β³−β².β⁴=1) such thatπ_(α)(β¹,β²,β³,β⁴)εGL₄(Z[i])

Proposition 8:

Let μ be a fundamental unit of Z[i,θ] with |μ|>1. Let (γ¹,γ²)εZ[i,θ]²,then (β¹,β²)εZ[i,θ]² exists such that:

${{{\overset{\sim}{\Phi}}_{1}}{\left( {\beta^{1},\beta^{2}} \right) \cdot {{\overset{\sim}{\Phi}}_{2}}}\left( {{\overset{\_}{\beta}}^{1},{\overset{\_}{\beta}}^{2}} \right)} = {{{\overset{\sim}{\Phi}}_{1}}{\left( {Y^{1},Y^{2}} \right) \cdot {{\overset{\sim}{\Phi}}_{2}}}\left( {{\overset{\_}{Y}}^{1},{\overset{\_}{Y}}^{2}} \right)}$and${{{{\overset{\sim}{\Phi}}_{1}}^{2}\left( {\beta^{1},\beta^{2}} \right)} + {{{\overset{\sim}{\Phi}}_{2}}^{2}\left( {{\overset{\_}{\beta}}^{1},{\overset{\_}{\beta}}^{2}} \right)}} \leq {{{\overset{\sim}{\Phi}}_{1}}{\left( {\beta^{1},\beta^{2}} \right) \cdot {{\overset{\sim}{\Phi}}_{2}}}{\left( {{\overset{\_}{\beta}}^{1},{\overset{\_}{\beta}}^{2}} \right) \cdot \left( {{\mu } + \frac{1}{\mu }} \right)}}$

Proof:

${{Let}\mspace{14mu} k} = {\left\lfloor {\frac{{\ln\left( {{{\overset{\sim}{\Phi}}_{1}}\left( {Y^{1},Y^{2}} \right)} \right)} - {\ln\left( {{{\overset{\sim}{\Phi}}_{2}}\left( {{\overset{\_}{Y}}^{1},{\overset{\_}{Y}}^{2}} \right)} \right)}}{\ln\left( {\mu }^{2} \right)} + 0.5} \right\rfloor.}$

We take:

β¹ = μ^(−k)Y¹et β² = μ^(−k)Y² Then  we  have${{{\overset{\sim}{\Phi}}_{1}}{\left( {\beta^{1},\beta^{2}} \right) \cdot {{\overset{\sim}{\Phi}}_{2}}}\left( {{\overset{\_}{\beta}}^{1},{\overset{\_}{\beta}}^{2}} \right)} = {{{\overset{\sim}{\Phi}}_{1}}{\left( {Y^{1},Y^{2}} \right) \cdot {{\overset{\sim}{\Phi}}_{2}}}\left( {{\overset{\_}{Y}}^{1},{\overset{\_}{Y}}^{2}} \right)}$${{and}\mspace{14mu}\frac{{Max}\left( {{{{\overset{\sim}{\Phi}}_{1}}\left( {\beta^{1},\beta^{2}} \right)},{{{\overset{\sim}{\Phi}}_{2}}\left( {{\overset{\_}{\beta}}^{1},{\overset{\_}{\beta}}^{2}} \right)}} \right)}{{Min}\left( {{{{\overset{\sim}{\Phi}}_{1}}\left( {\beta^{1},\beta^{2}} \right)},{{{\overset{\sim}{\Phi}}_{2}}\left( {{\overset{\_}{\beta}}^{1},{\overset{\_}{\beta}}^{2}} \right)}} \right)}} \leq {\mu }$

We then easily obtain:

${{{{\overset{\sim}{\Phi}}_{1}}^{2}\left( {\beta^{1},\beta^{2}} \right)} + {{{\overset{\sim}{\Phi}}_{2}}^{2}\left( {{\overset{\_}{\beta}}^{1},{\overset{\_}{\beta}}^{2}} \right)}} \leq {{{\overset{\sim}{\Phi}}_{1}}{\left( {\beta^{1},\beta^{2}} \right) \cdot {{\overset{\sim}{\Phi}}_{2}}}{\left( {{\overset{\_}{\beta}}^{1},{\overset{\_}{\beta}}^{2}} \right) \cdot \left( {{\mu } + \frac{1}{\mu }} \right)}}$

First step: balance between layers:

-   -   Search for (β¹,β²)εZ[i,θ]²

First of all, we search for the pair (β¹,β²)εZ[i,θ]² which minimizes∥{tilde over (φ)}₁∥.∥{tilde over (φ)}₂∥.

The determination of (β¹,β²)εZ[i,θ]² is equivalent to the determinationof the vector B of Z[i]⁴ defined by:

$B = {{{\cdot \begin{bmatrix}\beta_{1}^{1} \\\beta_{2}^{1} \\\beta_{1}^{2} \\\beta_{2}^{2}\end{bmatrix}}\mspace{14mu}{where}\mspace{14mu}\beta^{j}} = {\beta_{j}^{1} + {\theta \cdot \beta_{j}^{2}}}}$

By setting:

$\begin{matrix}{\Omega = \begin{bmatrix}\omega_{1} \\\omega_{2} \\\omega_{3} \\\omega_{4}\end{bmatrix}} \\{= {\underset{\underset{G_{\alpha,\theta}{(R)}}{︸}}{\begin{bmatrix}r_{1,1}^{1} & 0 & {i \cdot r_{1,2}^{1}} & 0 \\r_{2,1}^{1} & 0 & {i \cdot r_{2,2}^{1}} & 0 \\0 & r_{2,2}^{2} & 0 & r_{2,1}^{2} \\0 & r_{1,2}^{2} & 0 & r_{1,1}^{2}\end{bmatrix} \cdot \begin{bmatrix}1 & \theta & 0 & 0 \\1 & \overset{\_}{\theta} & 0 & 0 \\0 & 0 & \frac{\overset{\_}{\alpha}}{\alpha} & {\frac{\overset{\_}{\alpha}}{\alpha}\theta} \\0 & 0 & \frac{\alpha}{\overset{\_}{\alpha}} & {\frac{\alpha}{\overset{\_}{\alpha}}\overset{\_}{\theta}}\end{bmatrix}} \cdot \begin{bmatrix}\beta_{1}^{1} \\\beta_{2}^{1} \\\beta_{1}^{2} \\\beta_{2}^{2}\end{bmatrix}}} \\{= {{G_{\alpha,\theta}(R)} \cdot B}}\end{matrix}$

We therefore have:∥{tilde over (φ)}₁∥²=|ω₁|²+ω₂|²∥{tilde over (φ)}₂∥²=|ω₃|²+ω₄|²

The search algorithm runs as follows:

Initialization:

Search for the shortest vector of the lattice generated by the columnsof G_(α,θ)(R). B_(min) is initialized by the coordinates of one of theshortest vectors of this lattice:

$\left. B_{m\; i\; n}\leftarrow{{Arg}\left( {\underset{\underset{B \neq 0}{B \in {Z{\lbrack i\rbrack}}^{4}}}{Min}{{{G_{\alpha,\theta}(R)} \cdot B}}^{2}} \right)} \right.$

The value of ∥{tilde over (φ)}₁∥.∥{tilde over (φ)}₂∥ is calculated forB_(min) this value is annotated ρ.

Search in a List:

We prepare the list, annotated L, of the vectors of the latticegenerated by the columns of G_(α,θ)(R) which have a square norm lessthan or equal to

${\rho \cdot \left( {{\mu } + \frac{1}{\mu }} \right)}\mspace{14mu}{i.e.\text{:}}$$L = \left\{ {B \in {{{Z\lbrack i\rbrack}^{4}B} \neq {0\mspace{14mu}{such}\mspace{14mu}{that}\mspace{14mu}{{{G_{\alpha,\theta}(R)} \cdot B}}^{2}} \leq {\rho\left( {{\mu } + \frac{1}{\mu }} \right)}}} \right\}$

For each of the elements of the list, we calculate the value of ∥{tildeover (φ)}₁∥.∥{tilde over (φ)}₂∥, B_(min) is the element of this listwhich minimises ∥{tilde over (φ)}₁∥.∥{tilde over (φ)}₂∥.

Using Proposition 8, it is evident that the pair (β¹,β²)εZ[i,θ]²obtained using B_(min) minimises the product ∥{tilde over (φ)}₁∥.∥{tildeover (φ)}₂∥.

-   -   determination of (β³,β⁴)εZ[i,θ]²:

After having determined the pair (β¹,β²)εZ[i,θ]² which minimises theproduct ∥{tilde over (φ)}₁∥.∥{tilde over (φ)}₂∥, it suffices to resolvethe equation β¹.β³−β².β⁴=1, which has a solution according Proposition7, in order to determine the pair (β³,β⁴) Z[i,θ]². Such an equation isreduced to a system of 4 equations with 8 unknowns with coefficients inZ, in which solutions are sought in Z⁸. This type of system is easilyresolved by putting a rectangular matrix with integer coefficients inHNF form (see H Cohen).

-   -   Second step: balance within the layers:

On completion of the first step, we have:{tilde over (φ)}={tilde over (R)}.Δ _(α)(β¹,β²,β³,β⁴)

A QR decomposition of {tilde over (φ)} is carried out:{tilde over (φ)}=U.T

For each layer j (1≦j≦2), we define:

$k_{j} = \left\lfloor {\frac{{\ln\left( t_{{2 \cdot j},{2 \cdot j}} \right)} - {\ln\left( t_{{{{2 \cdot j} + 1},{{2 \cdot j} + 1}}\;} \right)}}{\ln\left( {\mu }^{2} \right)} + 0.5} \right\rfloor$

In this second step, we will use the reduction matrixΔ_(α)(μ^(−k) ¹ ,0,μ^(−k) ² ,0)

It will be noted that its associated matrix π_(α)(μ^(−k) ¹ ,0,μ^(−k) ²,0) is certainly in GL₄(Z[i]) thanks to Proposition 4 and to the factthat μ^(−k) ¹ ^(−k) ² is a unit and therefore |N_(Q[i,θ]/Q[i])(μ^(−k) ¹^(−k) ² )|=1

-   -   Conclusion of the reduction algorithm.

In conclusion, we define:{tilde over (ψ)}={tilde over (R)}.Δ _(α)(β¹,β²,β³,β⁴).Δ_(α)(μ^(−k) ¹,0,μ^(−k) ² ,0)={tilde over (R)}Δ _(α)(x ¹ ,x ² ,x ³ ,x ⁴)

with x¹=β¹.μ^(−k) ¹ , x²=β².μ^(−k) ¹ , x³=β³.μ^(−k) ² and x⁴=β⁴.μ^(−k) ²

The curve on the single FIGURE has the decoder performances for twovalues of p (p=1 and p=2) compared with those of the Maximum Likelihooddecoder and those of the MMSE-GDFE LLL ZF-DFE decoder. The performancesare expressed in PER (Packet Error Rate), depending in the Signal toNoise Ratio (SNR) in dB.

Of course, the invention is not limited to the examples which have justbeen described and numerous adjustments can be made to these exampleswithout exceeding the scope of the invention. The use of diamondmatrices instead of X matrices can be envisaged for basis reduction andfor decoding. Furthermore, the first step of the algebraic reductionalgorithm may also not be carried out or it can be modified to make iteasier to execute. However, to compensate for this drop in quality ofthe reduction, it will be necessary to take a fairly high value of p.

1. A method for decoding 2×2 space-time codes of a vector Y, inparticular of the Golden Code type, the decoding method comprising adecoder for performing the following steps: a filtering of the minimummean square error-generalized decision feedback equalizer (MMSE-GDFE)type is carried out by the decoder in order to obtain a vectorY^(F)=RX+W, with Y^(F) the vector Y filtered by means of a filter matrixF (“forward filter”), R a conventional matrix named “backward filter”, Xthe vector transmitted and W the complex noise matrix; a constellationrecentring is carried out by the decoder so that the vector to bedetermined becomes Z, such that Z=Y^(F)−2K_(M)RΓS₀=2K_(M)RΓS+W, K_(M)being the normalization constant for the constellation in question, Fbeing the Golden Code, and S the vector of the information symbols; astep of permutation of the elements of the matrices Z, R and F iscarried out by the decoder in order to obtain {tilde over (Z)}, {tildeover (R)} and {tilde over (Γ)} such that {tilde over (R)}^(H)R is amatrix in the form of an X and the complex number “i” is moved from Γ toR; a step of lattice basis reduction is carried out by the decoder suchthat {tilde over (Z)}=2K_(M){tilde over (Ψ)}{tilde over (Γ)}S′+W with{tilde over (Ψ)}={tilde over (R)}.Δ_(α)(x¹, x², x³, x⁴) and S′=π_(α)⁻¹(x¹, x², x³, x⁴).S, π_(α)(x¹, x², x³, x⁴) being a basis change matrix,π_(α)(x¹, x², x³, x⁴) is the matrix associated with Δ_(α)(x¹, x², x³,x⁴) which is a matrix in which the determinant module is equal to 1 anddefined such that Δ_(α) ^(H)Δ_(α) is a matrix in the form of an X; andthe zero forcing-decision feedback equalizer (ZF-DFE) decoding algorithmis applied by the decoder, in which among the four elements of twolayers are identified, each layer comprising the two elements of thediagonal or anti-diagonal of {tilde over (Z)}; the DFE algorithm isapplied layer by layer and the ZF algorithm within each of the layers.2. The method according to claim 1, characterized in that thepermutation of {tilde over (Z)} is as follows:$\overset{\sim}{Z} = \begin{bmatrix}z_{1,1} \\z_{2,2} \\z_{1,2} \\z_{2,1}\end{bmatrix}$ with the following convention $Z = {\begin{bmatrix}z_{1,1} \\z_{2,1} \\z_{1,2} \\z_{2,2}\end{bmatrix}.}$
 3. The method according to claim 2, characterized inthat {tilde over (R)} is a matrix in the form of an X such that:${\overset{\sim}{R} = {{\begin{bmatrix}r_{1,1}^{1} & 0 & 0 & {i \cdot r_{1,2}^{1}} \\0 & r_{2,2}^{2} & r_{2,1}^{2} & 0 \\0 & r_{1,2}^{2} & r_{1,1}^{2} & 0 \\r_{2,1}^{1} & 0 & 0 & {i \cdot r_{2,2}^{1}}\end{bmatrix}\mspace{14mu}{and}\mspace{14mu}\overset{\sim}{\Gamma}} = {\frac{1}{\sqrt{5}}\begin{bmatrix}\alpha & {\alpha\;\theta} & 0 & 0 \\\overset{\_}{\alpha} & {\overset{\_}{\alpha}\;\overset{\_}{\theta}} & 0 & 0 \\0 & 0 & \alpha & {\alpha\theta} \\0 & 0 & \overset{\_}{\alpha} & {\overset{\_}{\alpha}\;\overset{\_}{\theta}}\end{bmatrix}}}},$ α and θ being the Golden Code coefficients such that:${\theta = \frac{1 + \sqrt{5}}{2}},{\overset{\_}{\theta} = \frac{1 - \sqrt{5}}{2}},$α=1+i−i.θ, α=1+i−i. θ.
 4. The method according to claim 1 wherein:${\Delta_{\alpha}\left( {x^{1},x^{2},x^{3},x^{4}} \right)} = \begin{bmatrix}x^{1} & 0 & 0 & {\frac{\alpha}{\overset{\_}{\alpha}}x^{4}} \\0 & {\overset{\_}{x}}^{1} & {\frac{\overset{\_}{\alpha}}{\alpha}{\overset{\_}{x}}^{4}} & 0 \\0 & {\frac{\alpha}{\overset{\_}{\alpha}}{\overset{\_}{x}}^{2}} & {\overset{\_}{x}}^{3} & 0 \\{\frac{\overset{\_}{\alpha}}{\alpha}x^{2}} & 0 & 0 & x^{3}\end{bmatrix}$ and${\Pi_{\alpha}\left( {x^{1},x^{2},x^{3},x^{4}} \right)} = \begin{bmatrix}x_{1}^{1} & x_{2}^{1} & x_{1}^{4} & {x_{1}^{4} - x_{2}^{4}} \\x_{2}^{1} & {x_{1}^{1} + x_{2}^{1}} & x_{2}^{4} & {- x_{1}^{4}} \\{x_{1}^{2} + x_{2}^{2}} & {x_{1}^{2} + {2 \cdot x_{2}^{2}}} & {x_{1}^{3} + x_{2}^{3}} & {- x_{2}^{3}} \\{- x_{2}^{2}} & {{- x_{1}^{2}} - x_{2}^{2}} & {- x_{2}^{3}} & x_{1}^{3}\end{bmatrix}$ where ∀j 1≦j≦4 x^(i)=x₁ ^(j)+θ.x₂ ^(j) (x₁ ^(j) et x₂^(j)εZ[i]).
 5. The method according to claim 1, characterized in thatΔ_(α) is determined such that, during a QR decomposition, both elementsof a single layer are substantially of the same value.
 6. The methodaccording to claim 1, characterized in that {tilde over (Z)} is definedas two layers, a first layer consisting of elements on the diagonal anda second layer consisting of elements on the anti-diagonal, and in thatΔ_(α) is determined such that, during a QR decomposition, the product ofthe two elements of the second layer is greater than the product of thetwo elements of the first layer.
 7. The method according to claim 1,characterized in that {tilde over (Z)} is defined as two layers, a firstlayer composed of elements on the diagonal and a second layer composedof elements on the anti-diagonal, and in that during application of theZF algorithm for the second layer, p possible candidates are extracted,then for the first layer p possible candidates are also extracted andthen the best candidate vector is determined.
 8. The method according toclaim 7, characterized in that p is equal to
 2. 9. The method accordingto claim 1, characterized in that {tilde over (R)} and Δ_(α) are Xmatrices.
 10. The method according to claim 1, characterized in that{tilde over (R)} and Δ_(α) are diagonal matrices.